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Singularities of the Minimal Model Program by Sándor Kovács, János Kollár

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5

Semi-log canonical pairs

It was discovered by Kollár and Shepherd-Barron (1988) that in order to compactify the moduli theory of higher dimensional varieties we need stable pairs. That is, pairs (X, Δ) with semi-log canonical singularities and ample log canonical class KX + Δ. The aim of this chapter is to study these semi-log canonical singularities.

In general X is neither normal nor irreducible. Such varieties can be studied either using semi-log resolutions or by focusing on their normalization. Both of these approaches have difficulties.

A stable curve C has ordinary nodes, and we can encode C by giving a triple (, , τ) where is the ...

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